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首页> 外文期刊>Engineering analysis with boundary elements >The numerical solution of Fokker-Planck equation with radial basis functions (RBFs) based on the meshless technique of Kansa's approach and Galerkin method
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The numerical solution of Fokker-Planck equation with radial basis functions (RBFs) based on the meshless technique of Kansa's approach and Galerkin method

机译：基于Kansa方法和Galerkin方法的径向基函数（RBFs）的Fokker-Planck方程的数值解

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This paper describes two numerical methods based on radial basis functions (RBFs) for solving the time-dependent linear and nonlinear Fokker-Planck equations in two dimensions. These methods (RBFs) give a closed form for approximating the solution of partial differential equations. We approximate the linear and nonlinear Fokker-Planck equations with radial basis functions which are based on two techniques, one of them is Kansa's approach and another technique is the Galerkin method of Tau type [54]. In this work, we discretize the time variable with Crank-Nicolson method. For the space variable, we apply the radial basis functions which are Multiquadrics (MQ) and Inverse Quadric (IQ). Also, we employ another radial basis function which was introduced in. These basis functions depend on constant (shape) parameter. As is well known, the shape parameter has a strong influence on the accuracy of the numerical solutions and thus we test and compare several different strategies to choose this parameter. Both techniques (Kansa's approach and Tau method) yield a linear system of algebraic equations say AX=b. The matrix A is usually very ill-conditioned. We apply QR decomposition technique for solving the linear system arising from our approximations. Finally, some test problems are presented to illustrate the efficiency of the new methods for the numerical solution of linear and nonlinear Fokker-Planck equations. Also, to show the good accuracy of the method of radial basis functions, we compute the errors using L_∞, root mean square (RMS) and L_2 norms.
机译：本文介绍了两种基于径向基函数（RBF）的数值方法，用于求解二维时变线性和非线性Fokker-Planck方程。这些方法（RBF）给出了近似的偏微分方程解的封闭形式。我们使用基于两种技术的带有径向基函数的线性和非线性Fokker-Planck方程进行近似，其中一种是Kansa方法，另一种是Tau型Galerkin方法[54]。在这项工作中，我们使用Crank-Nicolson方法离散化时间变量。对于空间变量，我们应用径向基函数，即多二次方（MQ）和逆二次方（IQ）。此外，我们采用了引入的另一个径向基函数。这些基函数取决于常数（形状）参数。众所周知，形状参数对数值解的准确性有很大的影响，因此我们测试并比较了几种选择该参数的不同策略。两种技术（Kansa方法和Tau方法）都产生一个代数方程的线性系统，即AX = b。矩阵A通常情况非常恶劣。我们应用QR分解技术来求解近似值所产生的线性系统。最后，提出了一些测试问题，以说明用于线性和非线性Fokker-Planck方程数值解的新方法的有效性。另外，为了显示径向基函数方法的良好准确性，我们使用L_∞，均方根（RMS）和L_2范数来计算误差。

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